Here's an article talking about exactly what you are asking:
EEG decoding of spoken words in bilingual listeners: from words to language invariant semantic-conceptual representations
I can say that the brain obviously has the ability to invariantly represent different stimuli as being the same abstract concept, not even from stimuli in the same domain (such as auditory to auditory) but even across different domains of knowledge (identifying astract conceptual relationships between visual signals and motor signals)! Whether this can be derived purely from EEG, I doubt it because it lacks the precision necessary (which would require single cell recordings of millions of cells at the same time). I should also note that EEG can't tell us much on how the brain generates the spectral graphs that we record, just what the electrical activity is. We also don't even know what exactly causes the signals. Electrical activity may not even be the only indication of invariance, because other underlying mechanisms may be involved. But the idea of invariance, itself, is the very idea of what you are talking about and this phenomenon is found in lots of ways. In fact, anytime anyone relates two different stimuli as the same concept, it's clear that somehow the brain has an invariant representation of the event. There exists some kind of mechanism in the brain to recognize and store abstract concepts, removed from the concrete signals from which the concepts are learned.
"Mirror" neurons are an example of the domain crossing invariance, and something you can Google. These neurons demonstrate invariance by being similarly activated when an animal witnesses another animal performing an action and when the animal performs the action itself (e.g. eating).
However, your question about bilingual invariant representations and the mirror neurons are simply specific examples of a more general phenomenon. In fact, the ability for humans to have such incredible invariance is a major substrate that makes our intellectual capabilities so distinctly advanced compared to even the latest and greatest artificial intelligence attempts.
In the auditory language vs. language example you provided, the differences in auditory signals are ignored and our perception of the concept being communicated remains invariant. The same thing happens in vision. Even though the objective visual signals received by the brain are different, objects can be conceptually related. Just look at one face, then look at another one. You will still be able to identify both objects as the singular concept, "face". You might say "but the general structure of the face is still there". Yet, if you actually look at firing patterns of the neurons, they are still completely different. It's just as mysterious how the brain does this as it is that we can identify differing audio signals as the same concept. Humans possess a variety of invariant abilities in object recognition, such as scale invariance (different sizes), rotational invariance, translational invariance, skew invariance, color invariance, contrast invariance, and many others.
Just keep in mind that even though we possess these invariant abilities, we also maintain the distinctions in parallel. So the answer to your question is both yes and no. We generate both an invariant representation and a variant representation of whatever the stimulus is. Theoretically, both representation types are present in the EEG. However, EEG is not powerful enough to identify the specifics of how this works in language and whether this is even true.